Contact projective structures and chains

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ČAP Andreas ŽÁDNÍK Vojtěch

Year of publication 2010
Type Article in Periodical
Magazine / Source Geometriae Dedicata
MU Faculty or unit

Faculty of Education

Web DOI 10.1007/s10711-009-9426-6
Field General mathematics
Keywords Projective structure; Contact projective structure; Path geometry; Fefferman construction; Chains; Cartan connection; Parabolic geometry
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Description D.J. Fox associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox's construction in terms of the equivalent parabolic (Cartan) geometries, showing that it is an analog of Fefferman's construction of a conformal structure associated to a CR structure. We show that, on the level of Cartan connections, this Fefferman-type construction is compatible with normality if and only if the initial structure has vanishing contact torsion. This leads to a geometric description of the paths that have to be added to the contact geodesics of a contact projective structure in order to obtain the subordinate projective structure. They are exactly the chains associated to the contact projective structure, which are analogs of the Chern-Moser chains in CR geometry. Finally, we analyze the consequences for the geometry of chains and prove that a chain-preserving contactomorphism must be a morphism of contact projective structures.
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